MEANS OF EXPRESSING FINENESS OF GRAIN.
If all the particles of clay were considered as being spheres or cubes their superficial areas would be inversely proportioned to their diameters. The following calculations show this to be true in regard to the sphere : Volume of a sphere is equal to Pi —; then if D and d are the diameters 6 Pi Pi of two spheres their volumes would be proportional as 6 6.
The number of spheres required to equal in volume a standard unit Pi 6 6 volume would be 1 - or in the one case, and in the 6 Pi Pi other. Since the surface of a sphere of each size is equal respectively to Pi and Pi the total surface area of a collection of spheres, having a 6 6 total volume equal to unity, would be in each case - x and Pi Pi d3 X or and respectively. The combined areas of each group PiD Pid of spheres occupying the same volume but having different diameters are, therefore, inversely proportional to their diameters. This proportional relation of the surface of the particles in the several groups is taken as the surface factor of the respective groups, and the sum of these as the surface factor of the clay.
Cushman' has shown the error involved in thus taking the mean of the extreme diameters in a given group. According to data given by Cush man, a mechanical analysis of the separate groups would show a predom inance (77 to 87 per cent in Cushman data) of the finer particles of that group. That the mean diameter obtained as described above, is not a true
mean of the diameters of the particles in a group, is obvious. The error thus involved cannot, however, be obviated without a much more exten sive subdivision of the groups than is possible under ordinary conditions. It needs no mathematical demonstration to make clear that, theoretically, the more extensive the analysis, the more accurate would the results be. It needs but a short experience with the mechanical analysis by any of the hydraulic methods, to learn that, practically, the more extensive the analysis is made, the larger will be the operating errors. In making a mechanical analysis one must choose between the Scylla and Charybdis of these errors and, naturally, will decide in favor of that one which involves the making of the fewest determinations.
In this report the mean of the extreme diameters of each group, irre spective of the distribution by number according to their volume, of the particles within the respective groups is taken as representing the diam eter of the group. The mean diameter of each group and total surface factors for the clays here reported are shown in Table V.